When a cyclist rides a cycle the wheel of the cycle having radius 77 cm makes 500 revolutions in 5 minutes .Find the speed of the cycle in km per hour . (Take `pi=(22)/(7))`

Answer

To find the speed of the cycle in kilometers per hour, we can follow these steps:

### Step 1: Calculate the Circumference of the Wheel
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
where \( r \) is the radius of the wheel.

Given:
- Radius \( r = 77 \) cm
- \( \pi = \frac{22}{7} \)

Substituting the values:
\[ C = 2 \times \frac{22}{7} \times 77 \]

### Step 2: Simplify the Circumference Calculation
Calculating the above expression:
\[ C = 2 \times \frac{22}{7} \times 77 = \frac{44 \times 77}{7} \]
Calculating \( 44 \times 77 \):
\[ 44 \times 77 = 3388 \]
Now, substituting this back:
\[ C = \frac{3388}{7} = 484 \text{ cm} \]

### Step 3: Calculate the Total Distance Covered in 500 Revolutions
The distance covered in one revolution is equal to the circumference. Therefore, the total distance \( D \) covered in 500 revolutions is:
\[ D = 500 \times C = 500 \times 484 \]
Calculating this:
\[ D = 242000 \text{ cm} \]

### Step 4: Convert Distance from Centimeters to Kilometers
To convert centimeters to kilometers, we use the conversion factor \( 1 \text{ km} = 100000 \text{ cm} \):
\[ D = \frac{242000}{100000} = 2.42 \text{ km} \]

### Step 5: Convert Time from Minutes to Hours
The time given is 5 minutes. To convert this into hours:
\[ \text{Time in hours} = \frac{5}{60} = \frac{1}{12} \text{ hours} \]

### Step 6: Calculate the Speed of the Cycle
Speed \( S \) is given by the formula:
\[ S = \frac{D}{\text{Time}} \]
Substituting the values we have:
\[ S = \frac{2.42}{\frac{1}{12}} = 2.42 \times 12 = 29.04 \text{ km/h} \]

### Final Answer
The speed of the cycle is:
\[ \text{Speed} = 29.04 \text{ km/h} \]

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